Say I have two disks $A$ and $B$, with two points $a\in A$ and $b\in B$. Is there a way to explicitly construct a linear fractional transformation from $A$ onto $B$ that sends $a$ to $b$?

I know a linear fractional transformation is determined by its image on 3 distinct points, and that they sends circles and lines to circles and lines. Would it then be enough to map $a\mapsto b$, and then choose two boundary points on $A$ to map to two arbitrary boundary points on $B$, or does more care need to be taken? My worry is that the boundary points may map onto a circle which isn't the boundary of $B$.

1 Answer
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Define the two LFT's that take your two circles, with distinguished interior points $a,b$ each to the unit disk and $0.$ Then take the inverse of one of the maps (given by the LFT of the inverse matrix) and compose it with the other map, the result takes $a$ to $b$ or $b$ to $a$ depending on order.